higher algebra

universal algebra

Algebraic theories

• algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory

• monad / (∞,1)-monad

• operad / (∞,1)-operad

Algebras and modules

• algebra over a monad

  ∞-algebra over an (∞,1)-monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

• algebra over an operad

  ∞-algebra over an (∞,1)-operad

• action, ∞-action

• representation, ∞-representation

• module, ∞-module

• associated bundle, associated ∞-bundle

Higher algebras

• monoidal (∞,1)-category

  • symmetric monoidal (∞,1)-category

  • monoid in an (∞,1)-category

  • commutative monoid in an (∞,1)-category

• symmetric monoidal (∞,1)-category of spectra

  • smash product of spectra

  • symmetric monoidal smash product of spectra

  • ring spectrum, module spectrum, algebra spectrum

• A-∞ algebra

  • A-∞ ring, A-∞ space

• C-∞ algebra

• E-∞ ring, E-∞ algebra

  • ∞-module, (∞,1)-module bundle

  • multiplicative cohomology theory

• L-∞ algebra

  • deformation theory

Model category presentations

• model structure on simplicial T-algebras / homotopy T-algebra

• model structure on operads

  model structure on algebras over an operad

Geometry on formal duals of algebras

• Isbell duality

• derived geometry

Theorems

• Deligne conjecture

• delooping hypothesis

• monoidal Dold-Kan correspondence

(∞,1)-category theory

Background

• category theory

• higher category theory

• (n,r)-category

Basic concepts

• (∞,1)-category

• hom-objects

• equivalences in/of $(\infty,1)$-categories

• sub-(∞,1)-category

  • reflective sub-(∞,1)-category

  • reflective localization

• opposite (∞,1)-category

• over (∞,1)-category

  • join of quasi-categories

• (∞,1)-functor

  • exact (∞,1)-functor

  • (∞,1)-category of (∞,1)-functors

  • (∞,1)-category of (∞,1)-presheaves

• fibrations

  • inner fibration

  • left/right fibration

  • Cartesian fibration

    • Cartesian morphism

Universal constructions

• limit

  • terminal object

• adjoint functors

Local presentation

• locally presentable

  • essentially small

  • locally small

  • accessible

  • idempotent-complete

Theorems

• (∞,1)-Yoneda lemma

• (∞,1)-Grothendieck construction

• adjoint (∞,1)-functor theorem

• (∞,1)-monadicity theorem

Extra stuff, structure, properties

• stable (∞,1)-category

• (∞,1)-topos

Models

• category with weak equivalences

• model category

• derivator

• quasi-category

  • model structure for quasi-categories

  • model structure for Cartesian fibrations

• relation to simplicial categories

  • homotopy coherent nerve

  • simplicial model category

  • presentable quasi-category

• Kan complex

  • model structure for Kan complexes

monoid theory in algebra:

• monoid, infinity-monoid

• monoid object, monoid object in an (infinity,1)-category

  • semiring, rig, ring, associative unital algebra

• Mon, CMon

• monoid homomorphism

• trivial monoid

• submonoid, quotient monoid?

• divisor, multiple?, quotient element?

• inverse element, unit, irreducible element

• ideal in a monoid

• principal ideal in a monoid

• commutative monoid

  • tensor product of commutative monoids

• cancellative monoid

• GCD monoid

• unique factorization monoid

• Bézout monoid

• principal ideal monoid

• group, abelian group

• absorption monoid

  • zero divisor

  • integral monoid

• free monoid, free commutative monoid

• graphic monoid

• monoid action

• module over a monoid

• localization of a monoid

• group completion

• endomorphism monoid

• super commutative monoid